On the existence of a solution to stochastic Navier-Stokes equations

The existence of a martingal solution to the following system of stochastic Navier-Stokes equations: ∂tu-vΔu+〈u,▽u〉+▽p = F(t,u)+G(t,u)dW/dt, div u = 0 considered with the Direchlet boundary condition u|PTLO = 0 in case O≠Rd was studied. It is assumed that O is connected with the boundary ∂O of class C2. A bounded time interval [0,T] was fixed. u(t,x) and p(t,x) as the velocity and pressure of an incompressible fluid at point x at time t is interpreted. It is shown that the F(t,u)+G(t,u)dW/dt is the density of the force per unit volume. The force has the deterministic and random parts which depend on u.